Orthonormal basis
Specific linear basis (mathematics) / From Wikipedia, the free encyclopedia
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In mathematics, particularly linear algebra, an orthonormal basis for an inner product space with finite dimension is a basis for
whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.[1][2][3] For example, the standard basis for a Euclidean space
is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for
arises in this fashion.
For a general inner product space an orthonormal basis can be used to define normalized orthogonal coordinates on
Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of
under the dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process.
In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces.[4] Given a pre-Hilbert space an orthonormal basis for
is an orthonormal set of vectors with the property that every vector in
can be written as an infinite linear combination of the vectors in the basis. In this case, the orthonormal basis is sometimes called a Hilbert basis for
Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required.[5] Specifically, the linear span of the basis must be dense in
although not necessarily the entire space.
If we go on to Hilbert spaces, a non-orthonormal set of vectors having the same linear span as an orthonormal basis may not be a basis at all. For instance, any square-integrable function on the interval can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonormal basis), but not necessarily as an infinite sum of the monomials
A different generalisation is to pseudo-inner product spaces, finite-dimensional vector spaces equipped with a non-degenerate symmetric bilinear form known as the metric tensor. In such a basis, the metric takes the form
with
positive ones and
negative ones.